# Electronic Theory of the Magnetic Order, Anisotropy and Reversal Processes in Rolled-Up Stripes and Thin Films

A comprehensive multiscale approach consisting of a combination of state of the art, first principles quantum theory and the flexible statistical methodologies of complex systems is applied in order to achieve a detailed microscopic understanding of the magnetic properties of rolled-up transition-metal [TM] ultrathin films and stripes. Quantum electronic investigations of the effective interactions between the local magnetic moments is performed as a function of composition and rolling radius *R* including the infinite radius *R* of planar films as well as very small *R*. This includes a detailed quantification of the isotropic and anisotropic exchange couplings, antisymmetric Dzyaloshinsky-Moriya [DM] couplings and local anisotropies. In addition, the effect of curvature on the magnetic energy landscapes of the nanorolls is characterized.

Quantum electronic calculations including spin-orbit coupling [SOC] performed on two-dimensional [2D] planar L10 bilayers are analyzed and the effective interaction parameters **D**_{ij} , J^{u,v,w}_{ij} and K^{u,v,w}_{i} of a generalized classical Heisenberg model [GHM] between local magnetic moments *i* are derived. Thus, the origin of magnetocrystalline anisotropy energy and its dependence on the chirality of noncollinear arrangements of the local magnetic moments in spin-density waves [SDWs] is analyzed from a local point of view. Our density-functional-theory [DFT] calculations show that the easy axis of TM nanostructures is most often the direction along which the interactions among the local magnetic moments are strongest.

Fig. 1: (a) Frozen-magnon dispersion relations with and without SOC obtained in *ab initio* calculations (symbols) in the case of an FePt bilayer (L10). The lines connecting the symbols represent the fits to the generalized classical Heisenberg model. (b) Illustration of an Fe monolayer (ML) on top of a Pt ML as in the L1_{0} structure. The orange spheres correspond to Fe atoms and white spheres to the Pt atoms. The arrows at the atoms indicate the direction of the local magnetic moments. From left to right the spins are rotated clockwise (-).

The magnetic energy landscapes of planar and rolled-up TM bilayers are characterized as obtained in the framework of a GHM, whose interaction parameters J^{u,v,w}_{ij} , **D**_{ij} and K^{u,v,w}_{i} up to ninth NNs are derived from DFT calculations on the planar TM bilayers. Particular attention is thereby paid to the geometrical shape and lifetime of noncollinear metastable states such as skyrmions and the minimum energy paths [MEPs] connecting them. It is shown that the symmetry of the magnetic configurations affects the excitation energies of the metastable states and their transformations along the MEPs. In addition, it is demonstrated that the skyrmions exhibit a number of characteristic properties of quasiparticles. They can move almost freely and nearly barrierless across the bilayers and cylinders, and they do not seem to interact strongly with each other, even at relatively short distances.

Fig. 2: Representative magnetic configurations of excited energy levels of the FePt cylinder are shown in 3D representation (upper figures) and in 2D projected or unrolled form. The arrows indicate the directions of the local magnetic moments at the Fe atoms.